Topological aspects of suitable theories

Simmons, Harold

doi:10.1017/s001309150001049x

Cited by 11 publications

(10 citation statements)

References 2 publications

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“…Having defined the notion of Magari algebra, the first question one can ask is whether there are any other natural examples of such algebras apart from the provability algebras of the form (L T , ✸ T ). The fact that such algebras naturally emerge from scattered topological spaces was discovered independently by Harold Simmons [44] and Leo Esakia [27]. 1 We now infer this semantics from rather general considerations.…”

Section: Topological Interpretationmentioning

confidence: 65%

Topological Interpretations of Provability Logic

Beklemishev

Gabelaia

2014

Leo Esakia on Duality in Modal and Intuitionistic Logics

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Provability logic concerns the study of modality ✷ as provability in formal systems such as Peano arithmetic. Natural, albeit quite surprising, topological interpretation of provability logic has been found in the 1970's by Harold Simmons and Leo Esakia. They have observed that the dual ✸ modality, corresponding to consistency in the context of formal arithmetic, has all the basic properties of the topological derivative operator acting on a scattered space. The topic has become a long-term project for the Georgian school of logic led by Esakia, with occasional contributions from elsewhere.More recently, a new impetus came from the study of polymodal provability logic GLP that was known to be Kripke incomplete and, in general, to have a more complicated behavior than its unimodal counterpart. Topological semantics provided a better alternative to Kripke models in the sense that GLP was shown to be topologically complete. At the same time, new fascinating connections with set theory and large cardinals have emerged.We give a survey of the results on topological semantics of provability logic starting from first contributions by Esakia. However, a special emphasis is put on the recent work on topological models of polymodal provability logic. We also included a few results that have not been published so far, most notably the results of Section 6 (due the second author) and Sections 10, 11 (due to the first author).

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Section: Topological Interpretationmentioning

confidence: 65%

Topological Interpretations of Provability Logic

Beklemishev

Gabelaia

2014

Leo Esakia on Duality in Modal and Intuitionistic Logics

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“…A proof of this proposition builds upon the ideas of H. Simmons [13] and L. Esakia [9,10], which by now have become almost folklore, but it is somewhat lengthy. For the reader's convenience we give this proof in the Appendix.…”

Section: Axiomsmentioning

confidence: 95%

“…any class of Kripke frames. However, a more general topological semantics for the Gödel-Löb provability logic GL has been known since the work of Simmons [13] and Esakia [9]. In the sense of this semantics, the diamond modality is interpreted as the topological derivative operator acting on a scattered topological space.…”

Section: Introductionmentioning

confidence: 99%

Topological completeness of the provability logic GLP

Beklemishev

Gabelaia

2013

Annals of Pure and Applied Logic

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“…This class of structures gives rise to the logic GL. Theorem 3.7 (Simmons [23] and Esakia [7]). GL is the logic of all scattered topological derivative spaces, as well as the logic of all converse well-founded derivative frames and the logic of all finite, transitive, irreflexive derivative frames.…”

Section: Dynamic Topological Logicsmentioning

confidence: 99%

Dynamic Cantor Derivative Logic

Fernández–Duque¹,

Montacute²

2021

Preprint

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Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as d-logics. Unlike logics based on the topological closure operator, d-logics have not previously been studied in the framework of dynamical systems, which are pairs (X, f ) consisting of a topological space X equipped with a continuous function f : X → X.We introduce the logics wK4C, K4C and GLC and show that they all have the finite Kripke model property and are sound and complete with respect to the d-semantics in this dynamical setting. In particular, we prove that wK4C is the d-logic of all dynamic topological systems, K4C is the d-logic of all TD dynamic topological systems, and GLC is the d-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where f is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems wK4H, K4H and GLH.The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological d-logics. Furthermore, our result for GLC constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation -something known to be impossible over the class of all spaces.

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Topological aspects of suitable theories

Cited by 11 publications

References 2 publications

Topological Interpretations of Provability Logic

Topological Interpretations of Provability Logic

Topological completeness of the provability logic GLP

Dynamic Cantor Derivative Logic

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